Question

Find a sinusoidal function for the following graph.

Short answer

The sinusoidal function for given graph is :$y=3\cos \left(\frac{x}{2}\right)or3\sin \left(\frac{\mathrm{\pi }}{2}-\frac{x}{2}\right)$

Step-by-step solution

Step 1. Given information 

Step 2. Assume the graph equation   

This shown graph shows the characteristics of the cosine function, it is symmetric about the y-axis

so let's assume the graph like

$y=A\cos \left(\omega x\right)$

from figure

amplitude $\left|A\right|=3$

period $T=12$

Step 3. calculating the ω

We know that $T=\frac{2\mathrm{\pi }}{\omega }$

$$\begin{gathered} 12=\frac{2\mathrm{\pi }}{\omega } \\ \omega =\frac{\mathrm{\pi }}{6} \end{gathered}$$

so,the cosine function is,for amplitude 3 and$\omega =\frac{\mathrm{\pi }}{6}$is

$y=3\cos \left(\frac{\mathrm{\pi }}{6}x\right)or3\sin \left(\frac{\mathrm{\pi }}{2}-\frac{\mathrm{\pi }}{6}x\right)$